GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Jun 2019, 00:31 GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  The numbers D, N, and P are positive integers, such that D < N, and N

Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 55716
The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

6
29 00:00

Difficulty:   95% (hard)

Question Stats: 25% (02:58) correct 75% (02:31) wrong based on 388 sessions

HideShow timer Statistics

The numbers D, N, and P are positive integers, such that D < N, and N is not a power of D. Is D a prime number?

(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2

Kudos for a correct solution.

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 55716
The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

2
8
Bunuel wrote:
The numbers D, N, and P are positive integers, such that D < N, and N is not a power of D. Is D a prime number?

(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

This is a tricky one.

Statement #1: two kinds of numbers have exactly four factors: (a) products of two distinct prime numbers, and (b) cubes of prime numbers.

The product of two distinct prime numbers S and T would have factors {1, S, T, ST}. For example, the factors of 10 are (1, 2, 5, 10), and the factors of 21 are {1, 3, 7, 21}.

The cube of a prime number S would have as factors 1, S, S squared, and S cubed. For example, 8 has factors {1, 2, 4, 8} and 27 has factors {1, 3, 9, 27}.

We know N is not a power of D, so the second case is excluded. N must be the product of two distinct prime numbers. We know D < N, so of the four factors, D can’t be the product of the two prime numbers. D could be either of the prime number factors, or D could be 1, which is not a prime number. Because D could either be a prime number or 1, we cannot give a definitive answer to the question. This statement, alone and by itself, is not sufficient.

Statement #2: this is tricky. The first few plug-ins seem to reveal a pattern. Even if you sense a pattern, it’s important to remember that plugging in numbers alone is never enough to establish that a DS statement is sufficient. Here, if we persevered to one more plug-in, we would find the one that breaks the pattern.
P = 5 --> 3^5 + 2 = 243, which is not a prime.

That gives another answer to the prompt, so we know this statement is not sufficient.

To avoid a lot of plugging in, it’s also very good to know that in mathematics, prime numbers are notorious for not following any easy pattern. It is impossible to produce an algebraic formula that will always produce prime numbers. In fact, this is more than you need to know, but the hardest unsolved question in higher mathematics, the Riemann Hypothesis, concerns the pattern of prime numbers; mathematicians have been working on this since 1859, and no one has proven it yet. Suffice to say that no one-line algebraic formula is going to unlock the mystery of prime numbers!

Combined statements: according to the information in statement #1, either D = 1 or D is a prime number. Well, statement #2 excludes the possibility that D = 1, because that number cannot be written as two more than a power of 3. Therefore, D must be a prime number. We have a definitive answer to the prompt question. Combined, the statements are sufficient.

Attachment: ghdmpp_img25.png [ 6.41 KiB | Viewed 5274 times ]

_________________
General Discussion
Manager  Joined: 14 Sep 2014
Posts: 106
Concentration: Technology, Finance
WE: Analyst (Other)
Re: The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

4
I will say C. Please let me know if I am missing something.

(1) The four factors of N include 1, D, something, and N. Therefore, N is the product of D and something.
Test case 1: N = 6 so D is either 2 or 3 and therefore is prime.
Test case 2: N = 8 so D is 4 and not prime (D cannot be 2 because 8 is a power of 2, which violates the constraints.)

(2) Possible values for D include 5, 11, 29, 83, 245, ... etc.
We have prime and non-prime possibilities, so this is insufficient.

Combining the statements, D must be one of the prime values given in statement 2, otherwise N would have more than four factors.
Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

Bunuel wrote:
The numbers D, N, and P are positive integers, such that D < N, and N is not a power of D. Is D a prime number?

(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2

Kudos for a correct solution.

Option A:N has exactly four factors, and D is a factor of N : Not Sufficient
$$2^1 * 3^1$$ factors are 1,2,3,6 ; D can be 2 or 1 . Not sufficient.
$$19^1 * 2^1$$ factors are 1,2,19,38 ; D can be 19,2,1 . (note 1 is not prime)
Option B: D = (3^P) + 2 Not Sufficient
$$P=1; D=5 is prime ? Yes$$
$$P=5; D=245 is prime ? NO$$

A && B
from B we know that D is and ODD Integer and from A we have
D can be 19,2,1 . (note 1 is not prime) 19 is prime but 1 is not so still the statement is insufficient.

Manager  Joined: 14 Sep 2014
Posts: 106
Concentration: Technology, Finance
WE: Analyst (Other)
The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

1
Lucky2783 wrote:
D can be 19,2,1 . (note 1 is not prime) 19 is prime but 1 is not so still the statement is insufficient.

If D = 1, then 3^P = -1 and P is not a positive integer, violating the constraint given in the problem. (Same with D = 2 or 19)
Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

sterling19 wrote:
Lucky2783 wrote:
D can be 19,2,1 . (note 1 is not prime) 19 is prime but 1 is not so still the statement is insufficient.

If D = 1, then 3^P = -1 and P is not a positive integer, violating the constraint given in the problem. (Same with D = 2 or 19)

good catch !!
Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

Lucky2783 wrote:
sterling19 wrote:
Lucky2783 wrote:
D can be 19,2,1 . (note 1 is not prime) 19 is prime but 1 is not so still the statement is insufficient.

If D = 1, then 3^P = -1 and P is not a positive integer, violating the constraint given in the problem. (Same with D = 2 or 19)

good catch !!

Hi Bunuel,

We (me and @sterling19) have come up with number substitution solution for this question but it took me 2-3 mins to write the test cases ,
is there a better way to attack this question apart from the magoosh official solution ?
Manager  B
Joined: 09 Oct 2015
Posts: 237
The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

Bunuel
Bunuel wrote:
Bunuel wrote:
The numbers D, N, and P are positive integers, such that D < N, and N is not a power of D. Is D a prime number?

(1) N has exactly four factors, and D is a factor of N
(2) D = (3^P) + 2

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

This is a tricky one.

Statement #1: two kinds of numbers have exactly four factors: (a) products of two distinct prime numbers, and (b) cubes of prime numbers.

The product of two distinct prime numbers S and T would have factors {1, S, T, ST}. For example, the factors of 10 are (1, 2, 5, 10), and the factors of 21 are {1, 3, 7, 21}.

The cube of a prime number S would have as factors 1, S, S squared, and S cubed. For example, 8 has factors {1, 2, 4, 8} and 27 has factors {1, 3, 9, 27}.

We know N is not a power of D, so the second case is excluded. N must be the product of two distinct prime numbers. We know D < N, so of the four factors, D can’t be the product of the two prime numbers. D could be either of the prime number factors, or D could be 1, which is not a prime number. Because D could either be a prime number or 1, we cannot give a definitive answer to the question. This statement, alone and by itself, is not sufficient.

Statement #2: this is tricky. The first few plug-ins seem to reveal a pattern. Even if you sense a pattern, it’s important to remember that plugging in numbers alone is never enough to establish that a DS statement is sufficient. Here, if we persevered to one more plug-in, we would find the one that breaks the pattern.
P = 5 --> 3^5 + 2 = 243, which is not a prime.

That gives another answer to the prompt, so we know this statement is not sufficient.

To avoid a lot of plugging in, it’s also very good to know that in mathematics, prime numbers are notorious for not following any easy pattern. It is impossible to produce an algebraic formula that will always produce prime numbers. In fact, this is more than you need to know, but the hardest unsolved question in higher mathematics, the Riemann Hypothesis, concerns the pattern of prime numbers; mathematicians have been working on this since 1859, and no one has proven it yet. Suffice to say that no one-line algebraic formula is going to unlock the mystery of prime numbers!

Combined statements: according to the information in statement #1, either D = 1 or D is a prime number. Well, statement #2 excludes the possibility that D = 1, because that number cannot be written as two more than a power of 3. Therefore, D must be a prime number. We have a definitive answer to the prompt question. Combined, the statements are sufficient.

if N is a cube of a prime, n = 27
in this case we can have d as either 1 or 9, right?
why have we completely ignored the case of n being a cube of a prime no?
Non-Human User Joined: 09 Sep 2013
Posts: 11396
Re: The numbers D, N, and P are positive integers, such that D < N, and N  [#permalink]

Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: The numbers D, N, and P are positive integers, such that D < N, and N   [#permalink] 29 Jan 2019, 04:41
Display posts from previous: Sort by

The numbers D, N, and P are positive integers, such that D < N, and N  